If "like" is a preposition, then "like old rocks or unique autos" is a PP modifying the preceding noun phrase ("a collection"). I would consider that to be incorrect in this sentence on semantic grounds, and I suppose that that is what your professor thinks, too; "a collection" and "old rocks or unique autos" don't really seem to be similar things.
We can replace the PP with an adverbial clause:
A collection, as old rocks or unique autos [[do]], gives a person some individuality.
Now "as" functions as a subordinating conjunction introducing the adverbial clause. The semantics of this sentence are fine; both "a collection" and "old rocks or unique autos" can give a person some individuality.
It is very common to replace the subordinator "as" with "like" (although many prescriptivists deprecate that practice), so your original sentence could be considered correct if you interpreted "like" in that way (i.e., as a subordinator).